CC -- the class of all complex numbers

Description

In Macaulay2, complex numbers are represented as floating point numbers, and so are only approximate. The symbol ii represents the square root of -1 in many numeric contexts. A complex number is obtained by using the symbolic constant ii or the conversion functions toCC and numeric, in combination with real numbers (see RR). It is stored internally as a pair of arbitrary precision floating point real numbers, using the MPFR library.

i1 : z = 3-4*ii

o1 = 3-4*ii

o1 : CC (of precision 53)

i2 : z^5

o2 = -237.0000000000017+3115.999999999999*ii

o2 : CC (of precision 53)

i3 : 1/z

o3 = .12+.16*ii

o3 : CC (of precision 53)

i4 : +ii

o4 = ii

o4 : CC (of precision 53)

i5 : numeric_200 ii

o5 = ii

o5 : CC (of precision 200)

Complex numbers are ordered lexicographically, mingled with real numbers.

i6 : sort {1+ii,2+ii,1-ii,2-ii,1/2,2.1,7/5}

      1              7
o6 = {-, 1-ii, 1+ii, -, 2-ii, 2+ii, 2.1}
      2              5

o6 : List

The precision is measured in bits, is visible in the ring displayed on the second of each pair of output lines, and can be recovered using precision.

i7 : precision z

o7 = 53

For complex numbers, the functions class and ring yield different results. That allows numbers of various precisions to be used without creating a new ring for each precision.

i8 : class z

o8 = CC

o8 : InexactFieldFamily

i9 : ring z

o9 = CC
       53

o9 : ComplexField

A computation involving numbers of different precisions has a result with the minimal precision occurring. Numbers that appear alone on an output line are displayed with all their meaningful digits. (Specifying 100 bits of precision yields about 30 decimal digits of precision.)

i10 : 3p100+2p90e3*ii

o10 = 3+2000*ii

o10 : CC (of precision 90)

Numbers displayed inside more complicated objects are printed with the number of digits specified by printingPrecision.

i11 : printingPrecision

o11 = 6

i12 : x = {1/3.*ii,1/3p100*ii}

o12 = {.333333*ii, .333333*ii}

o12 : List

Use toExternalString to produce something that, when encountered as input, will reproduce exactly what you had before.

i13 : y = toExternalString x

o13 = {toCC(.0p53,.33333333333333331p53),toCC(.0p100
      ,.33333333333333333333333333333346p100)}

i14 : value y === x

o14 = true

Caveat

Currently, most transcendental functions are not implemented for complex arguments.

Functions and methods returning a complex number:

CC * CC -- see * -- a binary operator, usually used for multiplication
CC * QQ -- see * -- a binary operator, usually used for multiplication
CC * RR -- see * -- a binary operator, usually used for multiplication
CC * ZZ -- see * -- a binary operator, usually used for multiplication
QQ * CC -- see * -- a binary operator, usually used for multiplication
RR * CC -- see * -- a binary operator, usually used for multiplication
ZZ * CC -- see * -- a binary operator, usually used for multiplication
+ CC -- see + -- a unary or binary operator, usually used for addition
CC + CC -- see + -- a unary or binary operator, usually used for addition
CC + QQ -- see + -- a unary or binary operator, usually used for addition
CC + RR -- see + -- a unary or binary operator, usually used for addition
CC + ZZ -- see + -- a unary or binary operator, usually used for addition
QQ + CC -- see + -- a unary or binary operator, usually used for addition
RR + CC -- see + -- a unary or binary operator, usually used for addition
ZZ + CC -- see + -- a unary or binary operator, usually used for addition
- CC -- see - -- a unary or binary operator, usually used for negation or subtraction
CC - CC -- see - -- a unary or binary operator, usually used for negation or subtraction
CC - QQ -- see - -- a unary or binary operator, usually used for negation or subtraction
CC - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
CC - ZZ -- see - -- a unary or binary operator, usually used for negation or subtraction
QQ - CC -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - CC -- see - -- a unary or binary operator, usually used for negation or subtraction
ZZ - CC -- see - -- a unary or binary operator, usually used for negation or subtraction
CC / CC -- see / -- a binary operator, usually used for division
CC / QQ -- see / -- a binary operator, usually used for division
CC / RR -- see / -- a binary operator, usually used for division
CC / ZZ -- see / -- a binary operator, usually used for division
QQ / CC -- see / -- a binary operator, usually used for division
RR / CC -- see / -- a binary operator, usually used for division
ZZ / CC -- see / -- a binary operator, usually used for division
acos(CC) -- see acos -- arccosine
agm(CC,CC) -- see agm -- arithmetic-geometric mean
agm(CC,RR) -- see agm -- arithmetic-geometric mean
agm(RR,CC) -- see agm -- arithmetic-geometric mean
asin(CC) -- see asin -- arcsine
atan(CC) -- see atan -- compute the arctangent of a number
Beta(CC,CC) -- see Beta -- Beta function
Beta(CC,RR) -- see Beta -- Beta function
Beta(RR,CC) -- see Beta -- Beta function
cos(CC) -- see cos -- compute the cosine
cosh(CC) -- see cosh -- compute the hyperbolic cosine
cot(CC) -- see cot -- cotangent
coth(CC) -- see coth -- hyperbolic cotangent
csc(CC) -- see csc -- cosecant
csch(CC) -- see csch -- hyperbolic cosecant
Digamma(CC) -- see Digamma -- Digamma function
eint(CC) -- see eint -- exponential integral
erf(CC) -- see erf -- error function
erfc(CC) -- see erfc -- complementary error function
exp(CC) -- see exp -- exponential function
expm1(CC) -- see expm1 -- exponential minus 1
Gamma(CC) -- see Gamma -- Gamma function
Gamma(CC,CC) -- see Gamma -- Gamma function
Gamma(CC,RR) -- see Gamma -- Gamma function
Gamma(RR,CC) -- see Gamma -- Gamma function
log(CC) -- see log -- logarithm function
log(RR,CC) -- see log -- logarithm function
log(RR,RR) -- see log -- logarithm function
log1p(CC) -- see log1p -- logarithm of 1+x
regularizedBeta(CC,CC,CC) -- see regularizedBeta -- regularized beta function
regularizedBeta(CC,CC,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(CC,RR,CC) -- see regularizedBeta -- regularized beta function
regularizedBeta(CC,RR,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,CC,CC) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,CC,RR) -- see regularizedBeta -- regularized beta function
regularizedBeta(RR,RR,CC) -- see regularizedBeta -- regularized beta function
regularizedGamma(CC,CC) -- see regularizedGamma -- upper regularized gamma function
regularizedGamma(CC,RR) -- see regularizedGamma -- upper regularized gamma function
regularizedGamma(RR,CC) -- see regularizedGamma -- upper regularized gamma function
sec(CC) -- see sec -- secant
sech(CC) -- see sech -- hyperbolic secant
sin(CC) -- see sin -- compute the sine
sinh(CC) -- see sinh -- compute the hyperbolic sine
sqrt(CC) -- see sqrt -- square root function
sqrt(RR) -- see sqrt -- square root function
tan(CC) -- see tan -- compute the tangent
tanh(CC) -- see tanh -- compute the hyperbolic tangent
toCC(CC) -- see toCC -- convert to high-precision complex number
toCC(QQ) -- see toCC -- convert to high-precision complex number
toCC(RR) -- see toCC -- convert to high-precision complex number
toCC(RR,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,CC) -- see toCC -- convert to high-precision complex number
toCC(ZZ,QQ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,QQ,QQ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,QQ,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,QQ,ZZ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR,QQ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,RR,ZZ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,ZZ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,ZZ,QQ) -- see toCC -- convert to high-precision complex number
toCC(ZZ,ZZ,RR) -- see toCC -- convert to high-precision complex number
toCC(ZZ,ZZ,ZZ) -- see toCC -- convert to high-precision complex number
zeta(CC) -- see zeta -- Riemann zeta function

Methods that use a complex number:

CC % CC -- see % -- a binary operator, usually used for remainder and reduction
CC % QQ -- see % -- a binary operator, usually used for remainder and reduction
CC % RR -- see % -- a binary operator, usually used for remainder and reduction
CC % ZZ -- see % -- a binary operator, usually used for remainder and reduction
CC // CC -- see // -- a binary operator, usually used for quotient
CC // QQ -- see // -- a binary operator, usually used for quotient
CC // RR -- see // -- a binary operator, usually used for quotient
CC // ZZ -- see // -- a binary operator, usually used for quotient
abs(CC) -- see abs -- absolute value function
CC + InfiniteNumber (missing documentation)
CC - InfiniteNumber (missing documentation)
CC == CC (missing documentation)
CC == QQ (missing documentation)
CC == RR (missing documentation)
CC == ZZ (missing documentation)
CC ? InfiniteNumber (missing documentation)
conjugate(CC) -- see conjugate -- complex conjugate
format(CC) -- see format -- format a string or a real number
InfiniteNumber * CC (missing documentation)
InfiniteNumber + CC (missing documentation)
InfiniteNumber - CC (missing documentation)
InfiniteNumber / CC (missing documentation)
InfiniteNumber // CC (missing documentation)
InfiniteNumber ? CC (missing documentation)
isReal(CC) -- see isReal -- whether a number is real
CC << ZZ -- see left shift
lift(CC,type of QQ) -- see lift -- lift to another ring
lift(CC,type of RR_*) -- see lift -- lift to another ring
lift(CC,type of ZZ) -- see lift -- lift to another ring
liftable(CC,type of RR_*) (missing documentation)
lngamma(CC) -- see lngamma -- logarithm of the Gamma function
numeric(CC) -- see numeric -- convert to floating point
numeric(ZZ,CC) -- see numeric -- convert to floating point
QQ == CC (missing documentation)
CC >> ZZ -- see right shift
ring(CC) -- see ring -- get the associated ring of an object
round(CC) -- see round -- round a number
RR == CC (missing documentation)
size2(CC) -- see size2 -- number of binary digits to the left of the point
ZZ == CC (missing documentation)

For the programmer

The object CC is an inexact field family, with ancestor classes InexactNumber < Number < Thing.